Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,36)(27,32,39,47)(28,33,42,50)(29,34,30,35)(37,45,54,56)(38,46,43,51)(40,48,53,55)(41,49,44,52);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,32)(28,40)(29,38)(30,43)(31,39)(33,48)(34,46)(35,51)(36,47)(37,41)(42,53)(44,54)(45,49)(50,55)(52,56);(2,7)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,41)(30,44)(31,42)(32,45)(34,49)(35,52)(36,50)(38,53)(39,54)(40,43)(46,55)(47,56)(48,51);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,30)(26,34,36,35)(27,38,39,43)(28,41,42,44)(32,46,47,51)(33,49,50,52)(37,53,54,40)(45,55,56,48);(1,4,5,3)(2,8,7,6)(11,21)(15,18)(19,20)(23,24)(27,38)(29,30)(32,46)(34,35)(37,53)(39,43)(40,54)(41,44)(45,55)(47,51)(48,56)(49,52)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6627058);